Spin foam models for quantum gravity from lattice path integrals

Spin foam models for quantum gravity are derived from lattice path integrals. The setting involves variables from both lattice BF theory and Regge calculus. The action consists in a Regge action, which depends on areas, dihedral angles and includes the Immirzi parameter. In addition, a measure is inserted to ensure a consistent gluing of simplices, so that the amplitude is dominated by configurations that satisfy the parallel transport relations. We explicitly compute the path integral as a sum over spin foams for a generic measure. The Freidel-Krasnov and Engle-Pereira-Rovelli models correspond to a special choice of gluing. In this case, the equations of motion describe genuine geometries, where the constraints of area-angle Regge calculus are satisfied. Furthermore, the Immirzi parameter drops out of the on-shell action, and stationarity with respect to area variations requires spacetime geometry to be flat.

Figure 1

The evaluation of this spin network is the fusion coefficient. Each spin $l_t$, $i_{tv}^{+}$, and $i_{tv}^{-}$ intertwines four representations labeling the four faces of the tetrahedron $t$. To define it unambiguously, edges and vertices need to be oriented. The orientations have to be consistently chosen when summing the fusion coefficients with $\{15j\}$ symbols and other fusion coefficients.